Related papers: Poincar\'e duality and degrees of twisted Alexande…
We compute the adjoint twisted Reidemeister torsion for closed oriented hyperbolic $3$-manifolds and for hyperbolic $3$-manifolds with toroidal boundary. In our formula, we consider the manifold as obtained by doing a Dehn-filling along…
The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(\mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal…
Every element in the first cohomology group of a 3--manifold is dual to embedded surfaces. The Thurston norm measures the minimal `complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the…
Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring-spectrum and proved the Poincar\'e Duality theorem for this general case. In the current paper we consider oriented cohomology…
We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincar\'e duality in topology and Koszul duality in $\mathcal{E}_n$-algebra.
We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We…
We prove classical Taylor polynomial theorems for sub-Riemannian manifolds that are obtained as the submetric image of a Carnot group. For these theorems we also prove a sufficient condition for real analyticity and a result on…
We discuss the consequences of the Poincar\'e duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincar\'e duality property implies the existence of twisted…
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a…
The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.
We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic…
We prove a duality for factorization homology which generalizes both usual Poincar\'e duality for manifolds and Koszul duality for $\mathcal{E}_n$-algebras. The duality has application to the Hochschild homology of associative algebras and…
We present an accurate detailed exposition of the proof of existence of the Alexander-Conway polynomial (of links in 3-dimensional space). Other proofs were given by J. Alexander, J. Conway, V. Mantourov and L. Kauffman.
We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincar\'e duality for cohomology of tropical manifolds,…
In this note, we reformulate the invariant $\Delta (M, \omega)$ that we defined before, and show its relation with Reidemeister torsion. We calculate $\Delta (M, \omega)$ when the $3$-manifolds are lens spaces, and discuss the…
We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements provide proofs for the Manin-Mumford and the Bogomolov conjectures for Drinfeld…
An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple…
We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of…
The authors previously described an algebraic analogue of the JSJ-decomposition of a 3-manifold. This analogue is defined for any finitely presented, one-ended group. We study this analogue in the special case of Poincar\'e duality pairs.
We give an explicit formula of the Alexander polynomial of the link obtained by adding an arbitrary number of full twists to positively oriented parallel n-strands in terms of the Alexander polynomials of the links obtained by adding…